Following are the questions based on two statements and answer the following based on the given statements.
Side of square is 'a' cm. Find the value of 'a' ?
Statement I. Square is inscribed in a circle C1. Radius of circle C1 is 21cm.
Statement II Circle C2 is inscribed in the square. Radius of circle C2 is 28cm.

Options

A. Statement I alone is sufficient to answer the question while statement II alone is not sufficient to answer the question
B. Statement II alone is sufficient to answer the question while statement I alone is not sufficient to answer the question
C. Both statements I and II together are required to answer the question.
D. Either the statement I alone or Statement II alone is sufficient to answer the question

Answer

To find the value of 'a', the side of the square, we will analyze the two statements provided.

### Step 1: Analyze Statement I
**Statement I:** A square is inscribed in a circle C1 with a radius of 21 cm.

When a square is inscribed in a circle, the diameter of the circle is equal to the diagonal of the square. The relationship between the side length 'a' of the square and its diagonal can be expressed using the Pythagorean theorem:

\[ \text{Diagonal} = a\sqrt{2} \]

Since the radius of circle C1 is 21 cm, the diameter (which is twice the radius) is:

\[ \text{Diameter} = 2 \times 21 = 42 \text{ cm} \]

Setting the diagonal equal to the diameter:

\[ a\sqrt{2} = 42 \]

Now, we can solve for 'a':

\[ a = \frac{42}{\sqrt{2}} = 21\sqrt{2} \text{ cm} \]

### Step 2: Analyze Statement II
**Statement II:** A circle C2 is inscribed in the square with a radius of 28 cm.

When a circle is inscribed in a square, the radius of the circle is half the length of the side of the square. Therefore, if the radius of circle C2 is 28 cm, we can express this relationship as:

\[ \text{Radius} = \frac{a}{2} \]

Setting this equal to the given radius:

\[ \frac{a}{2} = 28 \]

Now, we can solve for 'a':

\[ a = 28 \times 2 = 56 \text{ cm} \]

### Conclusion
From Statement I, we found that \( a = 21\sqrt{2} \) cm, and from Statement II, we found that \( a = 56 \) cm.

Both statements are sufficient on their own to find the value of 'a', but they yield different results. However, since the question asks for the value of 'a', we can conclude that:

- **Statement I is sufficient to find 'a' as \( 21\sqrt{2} \) cm.**
- **Statement II is sufficient to find 'a' as 56 cm.**

Thus, the correct answer is that either statement alone is sufficient to answer the question.

### Final Answer
**Option 4:** Either statement I alone or statement II alone is sufficient to answer the question.

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